Changeset 10419 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_conservation.tex

Timestamp:

2018-12-19T20:46:30+01:00 (5 years ago)

Author:

smasson

Message:

dev_r10164_HPC09_ESIWACE_PREP_MERGE: merge with trunk@10418, see #2133

Location:

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex

Files:

• . (modified) (1 prop)
• NEMO (modified) (1 prop)
• NEMO/subfiles (modified) (1 prop)
• NEMO/subfiles/chap_conservation.tex (modified) (9 diffs)

NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_conservation.tex

@@ -1 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 
@@ -61 +62 @@
 Let us define as either the relative, planetary and total potential vorticity, i.e. ?, ?, and ?, respectively.
 The continuous formulation of the vorticity term satisfies following integral constraints:
-\begin{equation} \label{eq:vor_vorticity}
-\int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 
-\;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0
-\end{equation}
-
-\begin{equation} \label{eq:vor_enstrophy}
-if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 
-\frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} 
-=0
-\end{equation}
-
-\begin{equation} \label{eq:vor_energy}
-\int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
-\end{equation}
+\[
+  % \label{eq:vor_vorticity}
+  \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma
+        \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0
+\]
+
+\[
+  % \label{eq:vor_enstrophy}
+  if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot
+    \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv}
+  =0
+\]
+
+\[
+  % \label{eq:vor_energy}
+  \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
+\]
 where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element.
 (II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part.
@@ -122 +126 @@
 potential energy due to buoyancy forces:
 
-\begin{equation} \label{eq:hpg_pe}
-\int_D {-\frac{1}{\rho _o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv}
-\end{equation}
+\[
+  % \label{eq:hpg_pe}
+  \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv}
+\]
 
 Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of
@@ -142 +147 @@
 In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of 
 surface pressure forces is exactly zero:
-\begin{equation} \label{eq:spg}
-\int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0
-\end{equation}
+\[
+  % \label{eq:spg}
+  \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0
+\]
 
 (II.4.4) is satisfied in discrete form only if
@@ -159 +165 @@
 In continuous formulation, the advective terms of the tracer equations conserve the tracer content and
 the quadratic form of the tracer, $i.e.$
-\begin{equation} \label{eq:tra_tra2}
-\int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
-\;\text{and}
-\int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
-\end{equation}
+\[
+  % \label{eq:tra_tra2}
+  \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+  \;\text{and}
+  \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+\]
 
 The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies
@@ -180 +187 @@
 
 The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~:
-\begin{equation} \label{eq:dynldf_dyn}
-\int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla 
-_h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta 
-\;{\rm {\bf k}}} \right)} \right]\;dv} =0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_div}
-\int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
-\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
-\right]\;dv} =0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_curl}
-\int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
-\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
-\right]\;dv} \leqslant 0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_curl2}
-\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot 
-\nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h 
-\times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} 
-\leqslant 0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_div2}
-\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 
-{\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( 
-{A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0
-\end{equation}
+\[
+  % \label{eq:dynldf_dyn}
+  \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla
+        _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta
+            \;{\rm {\bf k}}} \right)} \right]\;dv} =0
+\]
+
+\[
+  % \label{eq:dynldf_div}
+  \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
+        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}
+    \right]\;dv} =0
+\]
+
+\[
+  % \label{eq:dynldf_curl}
+  \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
+        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}
+    \right]\;dv} \leqslant 0
+\]
+
+\[
+  % \label{eq:dynldf_curl2}
+  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot
+    \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h
+        \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv}
+  \leqslant 0
+\]
+
+\[
+  % \label{eq:dynldf_div2}
+  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[
+      {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left(
+          {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0
+\]
 
 
@@ -237 +249 @@
 conservation of momentum, dissipation of horizontal kinetic energy
 
-\begin{equation} \label{eq:dynzdf_dyn}
-\begin{aligned}
-& \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ 
-& \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\ 
- \end{aligned} 
- \end{equation}
+\[
+  % \label{eq:dynzdf_dyn}
+  \begin{aligned}
+    & \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\
+    & \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\
+  \end{aligned}
+\]
 conservation of vorticity, dissipation of enstrophy
-\begin{equation} \label{eq:dynzdf_vor}
-\begin{aligned}
-& \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
-}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
-{\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ 
-& \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
-}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
-{\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:dynzdf_vor}
+  \begin{aligned}
+    & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3
+          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm
+                  {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\
+    & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3
+          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm
+                  {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\
+  \end{aligned}
+\]
 conservation of horizontal divergence, dissipation of square of the horizontal divergence
-\begin{equation} \label{eq:dynzdf_div}
-\begin{aligned}
- &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
-k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
-\right)} \right)\;dv} =0 \\ 
-& \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
-k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
-\right)} \right)\;dv} \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:dynzdf_div}
+  \begin{aligned}
+    &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
+            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}
+          \right)} \right)\;dv} =0 \\
+    & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
+            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}
+          \right)} \right)\;dv} \leq 0 \\
+  \end{aligned}
+\]
 
 In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C).
@@ -286 +301 @@
 variance, i.e.
 
-\begin{equation} \label{eq:traldf_t_t2}
-\begin{aligned}
-&\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ 
-&\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:traldf_t_t2}
+  \begin{aligned}
+    &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\
+    &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\
+  \end{aligned}
+\]
 
 \textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, $i.e.$
 
-\begin{equation} \label{eq:trazdf_t_t2}
-\begin{aligned}
-& \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\ 
-& \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:trazdf_t_t2}
+  \begin{aligned}
+    & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\
+    & \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\
+  \end{aligned}
+\]
 
 Using the symmetry or anti-symmetry properties of the mean and difference operators,
@@ -311 +328 @@
 It has not been implemented.
 
+\biblio
+
 \end{document}